To accustom myself to the imminent mathematics course which will be taught in English totally in the university and practice my English writing,I will write latter articles in ,because I'm only a prospective college student,maybe there will be lots of mistakes and error concerning grammar,words and sentences,please don't mind.

【资料图】

In this article,we will study the Cauchy Integral theorem deeperly.

(the original edition of Cauchy theorem)

If a function  is analytic on a single-connective domain and there is a curve  which is a piecewisely smooth simple closed curve and is included entirely by the domain,then we have

Virtually,because a curve which is not simple can be divided into several simple curves by disconnecting its junctions,we can cancel the condition "simple" in the theorem.

From another aspect,since we said that a function is analytic on a closed domain means it is analytic on a domain including this closed domain,we can change the theorem into a version as follows:

(a better edition)

If a function  is analytic on a closed  domain

and  is a piecewisely smooth simple closed curve,then we have

Actually we have a ultimate version of the theorem concerning the single-connective domain.

(ultimate edition)

There is a curve  which is piecewisely smooth and simple closed with interior .Function  is analytic on  and is continuous on 

then we have

Theorem on complex-connected domain

Consider

which are piecewisely smooth simple closed  them,

are all included in the exterior of each other,but them are also all included in the interior of curve .Then we say that the point set in and simultaneously out of 

is a complex-connected domainwith 

as its boundary:

If an observer move along the boundary orientating its positive direction,points of the point set will always be on the observer's as

Under this circumstance,if function  is analytic on  and is continuous on

then we have

or

or

It tells us that the integral along the external boundary equals the sum of the integral along the internal boundaries.

If we want to prove this ,we just need to split the domain into two single-connective domains using line segments(or arcs)

And use the former theorem concerning single-connective that the integrals along the line segments(or arcs) can be offset,then we can find that the integral along zhe whole complex curve equals the integrals along the boundaries of the two single-connective 's zero.